Queueing Theory 2. Nikolaos Limnios

      COROLLARY 1.3.– Assume that X₂ is a strongly regenerative flow, B₀ satisfies [1.14] and

Then

If additionally Bi satisfies [1.14] and ρ₁ < 1, then Q₁(t) is a stable process.

      The first statement follows from corollary 1.1 since

is a strongly regenerative process. To prove the second statement, we note that Q₂ is a regenerative process and its stochastic boundedness means stability.

1.9. Queueing system with simultaneous service of a customer by a random number of servers

      Here, we consider a system S assuming that the nth customer requires service from ζn server simultaneously

The customer arrived in an empty queue begins service immediately if the number of available servers is more or equal ζn. Otherwise the customer becomes the first in a queue and the service begins when the required number of servers becomes available. A customer who arrives in an nonempty queue takes the last place in the queue. When service begins, each server’s completion time is independent of all other servers and has an exponential distribution with rate μ. The sequence

consists of iid random variables and

.

      Queueing systems with simultaneous service have been studied in a number of works (Rumyantsev and Morozov 2017; and references therein). The stability conditions in an explicit form have been obtained for systems with a Poisson input flow and independent exponentially distributed service times by (Gillent and Latouche 1983). The main goal of this section is an extension of the stability condition to the model with a regenerative input flow X based on theorems 1.1 and 1.2. Thus, we consider the system S described in section 1.2 with

is a sequence of independent exponentially distributed random variables not depending on ζn. Let S₀ be an auxiliary system defined in section 1.3. Instead of the process Y we consider the auxiliary flow Z that is the number of service completions by all m servers up to time t in the auxiliary system S₀. Denote by U(t) the number of occupied servers at time t in the system S₀. Then U is a Markov chain and Z is a doubly stochastic Poisson process (Grandell 1976) with a random intensity

We note that the process U hits the state {m} from any state j = 1, 2,..., m with a positive probability. It means that all states attainable from the state {m} constitute the finite class K of communicating states. Therefore, there are limits

Let

and

      We have the system of equations for

[1.18]

where

We may easily verify that the solution of [1.18] has the form

      Since

      we get

      and the traffic rate for the system S

      [1.19]

Let us note that this formula is the same as obtained by (Gillent and Latouche 1983) for queueing systems with a Poisson input flow. To employ theorems 1.1 and 1.2, we have to verify conditions 1.1, 1.3, 1.4 and 1.5. First of all we note that Z is a strongly regenerative flow. As points of regeneration, we may take the sequential hitting times

into the fixed state

(We take

if

Then

is the sojourn time in the state j and

is the return time to this state after exit from it for U(t). The random variables

has an exponential distribution. Moreover,

therefore, condition 1.1 holds. Let q(t) be the total number of servers that are already busy or will be busy by service of the Q(t) customers, which are present at the system S at time t. Then q(t) as well as Q(t) is a regenerative process with a sequence

of points of regeneration that is a subsequence of

such that

Let us recall that

is a sequence of points of regeneration for X. Therefore, because of theorem 1 from (Afanasyeva and Tkachenko 2014), condition 1.4 is fulfilled. Now for any fix

we define the common points of regeneration

for the input flow X and auxiliary flow Z by the relation

      Let

[1.20]